CalcuViz: Interactive Calculus with Python

Disclaimer: I am not a professional mathematician. CalcuViz is designed as a tool for visualization and understanding. For rigorous mathematical studies, please consult relevant academic sources.

CalcuViz is a Jupyter Notebook-based tool designed to offer an interactive environment for visualizing and understanding various calculus operations. It aims to bridge the gap between theory and its real-world applications.

Features:

1. Function Plotting:

• Theory: Visual representation of mathematical functions provides insight into their behavior. Given a function $f(x)$, the plot shows all points $(x, f(x))$ in the coordinate plane.

• Example Applications:

  • Analyzing revenue vs. cost functions in economics.
  • Observing the trajectory of a projectile in physics.

2. Derivative Operations:

• Theory: The derivative of a function represents its rate of change or the slope of the tangent line at any point. $f'(x) = \lim_{{h \to 0}} \frac{f(x+h) - f(x)}{h}$

• Example Applications:

  • Determining velocity from a position-time graph.
  • Understanding when a company's profits are increasing or decreasing the fastest.

3. Numerical Differentiation & Integration:

• Theory: Numerical methods offer approximations that can be useful in complex scenarios where symbolic methods are cumbersome. Forward difference: $f'(x) \approx \frac{f(x+h) - f(x)}{h}$

• Example Applications:

  • Calculating approximate changes in variables in engineering simulations.
  • Evaluating complex integrals in statistics.

4. Visualizing Integrals:

• Theory: Integration represents the area under the curve of a function, providing accumulative quantities. $\int f(x) dx$

• Example Applications:

  • Finding the total distance traveled using a velocity-time graph.
  • Computing the total energy consumption over a period.

5. Optimization Tasks:

• Theory: Optimization in calculus involves finding maximum or minimum values of functions. Local maxima or minima occur where the derivative is zero (or undefined) and the second derivative changes sign.

• Example Applications:

  • Determining the optimal pricing to maximize profit in business.
  • Engineering designs that require optimizing a particular parameter, like maximizing the strength of a beam with a given amount of material.

6. Taylor Series Expansion:

• Theory: The Taylor series provides polynomial approximations of functions about a specific point. The more terms in the series, the closer the approximation is to the original function within a certain range. $f(x) = f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \ldots$

• Example Applications:

  • Simplifying complex functions for easier analysis in physics or engineering.
  • Predicting future values of financial instruments using known data.

7. Root Finding:

• Theory: Root finding involves determining the values of $x$ for which $f(x) = 0$. Methods like Newton-Raphson provide iterative approaches to hone in on these root values.

• Example Applications:

  • Determining break-even points in business financial models.
  • Solving for equilibrium points in dynamic systems in engineering.

8. Solving Differential Equations using Euler's Method

Theory:
Differential equations involve functions and their derivatives, expressing relationships between varying quantities. Ordinary Differential Equations (ODEs) have a single unknown function and its derivatives.

Usage:

1. Define Your ODE: Create a Python function f(t, y) where t is the independent variable and y is the dependent variable.
2. Initial Condition: Specify the initial value of y as y0.
3. Time Parameters: Define the initial time t0, end time tn, and the step size h.

To solve your differential equation, run the following Python code:

solve_ode_euler(YOUR_ODE_FUNCTION, INITIAL_CONDITION, INITIAL_TIME, END_TIME, STEP_SIZE)

Replace placeholders with actual values or functions.

Example:

To solve $( \frac{dy}{dt} = y - t )$ with initial condition $( y(0) = 1 )$ from $( t = 0 )$ to $( t = 5 )$:

f = lambda t, y: y - t
solve_ode_euler(f, 1, 0, 5, 0.1)

Example Applications:

• Modeling the growth or decay of populations in biology.
• Describing the behavior of electrical circuits in engineering.

How to Use:

Option 1: Interactive Online Version with Binder

1. Click on the Binder badge:
   • This will open the notebook in an interactive environment directly in your browser.
2. Wait for the environment to load and initialize.
3. Interact with the notebook: input functions, select operations, and view the results!

Option 2: Local Setup

1. Clone the repository:

   git clone https://github.com/ElRapt/CalcuViz.git

2. Navigate to the repository's directory:

   cd CalcuViz

3. Set up a virtual environment and install the required packages:

   python -m venv calcuviz-env
   source calcuviz-env/bin/activate  # On Windows, use: .\calcuviz-env\Scripts\activate
   pip install -r requirements.txt

4. Start Jupyter Notebook:

   jupyter notebook

5. In the opened browser tab, navigate to calcuviz.ipynb and open it.

Feedback and Contributions:

Your feedback is invaluable! If you have suggestions, encounter bugs, or want to contribute, please open an issue or submit a pull request.